Hierarchical and ultrametric barriers in the energy landscape of jammed granular matter

Abstract

According to the mean-field glass theory, the (free) energy landscape of disordered systems is hierarchical and ultrametric if they belong to the full-replica-symmetry-breaking universality class. However, examining this theoretical picture in three-dimensional systems remains challenging, where the energy barriers become finite. Here, we numerically explore the energy landscape of granular models near the jamming transition using a saddle dynamics algorithm to locate both local energy minima and saddles. The multi-scale distances and energy barriers between minima are characterized by two metrics, both of which exhibit signatures of an ultrametric space. The scale-free distribution of energy barriers reveals that the landscape is hierarchical.

Figures (4)

• Figure 1: Energy landscape and pathway map. (a) A schematic multi-scale energy landscape. The inter-minima distance $d$ and barrier height $\Delta E$ both indicate the multi-scale nature of the landscape. (b) The pathway map of a two-scale, two-dimensional energy surface $E(x,y)$. The expression of $E(x,y)$ is shown in Supplemental Material (SM) Sec. S1. (c) Tree representation of the pathway map. We only show the stationary points in the three red dashed boxes.
• Figure 2: Statistical properties of distances and energy barriers measured from the native pathway map (without reconstruction). (a) The deviation of $d$ from perfect ultrametricity, $\mathcal{D}(\tilde{p})$. Data are obtained for particles with Harmonic interactions. (b) The probability density $f(E_{01})$ follows a power law with an exponent of about −2/3. (c) The relation between $E_{01}$ and $d_{01}$ reveals a power law with an exponent of about 4. (d) The probability density of $d_{01}$ compared to $f(d_{01}) \sim d_{01}^{1/3}$. Data in (b-d) are obtained for Hertzian systems with $N=64$ particles.
• Figure 3: Reconstruction of the pathway map with minimum energy pathways. (a) A schematic example of finding the min-max pathway $L_{\rm max}^{\rm min}(1,8)$ between two minima (1,8) from the native pathway map (from left to right). The matrix and tree representations of (b) $\Delta E$ and (c) $d^{<}$ on a set of local minima, organized using the clustering algorithm. The color scales with the corresponding value.
• Figure 4: Statistical properties of $\Delta E$. (a) Data of $P_>(\Delta E_{\rm c})$ and (inset) $f(\Delta E)$, for systems with $N=64$ particles. (b) Correspondence between $\Delta E_{\rm c}$ and $d_{\rm c}$.